Stable cheapest nonconforming finite elements for the Stokes equations

Abstract

We introduce two pairs of stable cheapest nonconforming finite element space pairs to approximate the Stokes equations. One pair has each component of its velocity field to be approximated by the $P_1$ nonconforming quadrilateral element while the pressure field is approximated by the piecewise constant function with globally two-dimensional subspaces removed: one removed space is due to the integral mean--zero property and the other space consists of global checker--board patterns. The other pair consists of the velocity space as the $P_1$ nonconforming quadrilateral element enriched by a globally one--dimensional macro bubble function space based on $DSSY$ (Douglas-Santos-Sheen-Ye) nonconforming finite element space; the pressure field is approximated by the piecewise constant function with mean--zero space eliminated. We show that two element pairs satisfy the discrete inf-sup condition uniformly. And we investigate the relationship between them. Several numerical examples are shown to confirm the efficiency and reliability of the proposed methods

P 1-nonconforming divergence-free finite element method on square meshes for Stokes equations

“…for the locally divergence-free finite element space V h in (13). Thus, to get u h in (26), we can solve the following elliptic problem for velocity:…”

“…The space N C consists of functions which are linear in each square and continuous on each midpoint of edge [14,15]. Recently, it has been proved that [N C] 2 is stable to solve Stokes equations with the piecewise constant space for velocity and pressure, respectively [13].…”

Nonconforming Finite Element Method Applied to the Driven Cavity Problem

“…The stability and optimal convergence properties of the two pairs of Stokes elements (5) and (6) are shown for the stationary Stokes equations in [28].…”

“…As the nonconforming elements use the values at the midpoints of edges as DOFs, instead of those at the vertices, the discontinuity singularities at the corners are naturally treated without any regularization. Our nonconforming finite element pairs are based on the two stable nonconforming finite element pairs on uniform square meshes [28] introduced for the stationary incompressible Stokes problem. The two pairs are briefly described as follows: The first of them uses the P 1 -nonconforming quadrilateral element [30] for the approximation of the velocity field, componentwise, while the pressure is approximated by a subspace of the piecewise constant functions whose dimension is two less than the number of squares in the mesh.…”

“…In order to treat the inhomogeneous lid-driven Dirichlet boundary condition, we modified the above elements [28] as follows. The boundary condition on the interior of the top boundary is handled as usual, but the corner boundary condition is specially treated at the two end elements on the top by adding two local DSSY-type bubble functions whose values at the midpoints of top boundary parts to be (1, 0) t and at the midpoint of the other boundary parts to be (0, 0) t .…”